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## Über dieses Buch

This book describes the basic principles of plasticity for students and engineers who wish to perform plasticity analyses in their professional lives, and provides an introduction to the application of plasticity theories and basic continuum mechanics in metal forming processes.

This book consists of three parts. The first part deals with the characteristics of plasticity and instability under simple tension or compression and plasticity in beam bending and torsion. The second part is designed to provide the basic principles of continuum mechanics, and the last part presents an extension of one-dimensional plasticity to general three-dimensional laws based on the fundamentals of continuum mechanics. Though most parts of the book are written in the context of general plasticity, the last two chapters are specifically devoted to sheet metal forming applications. The homework problems included are designed to reinforce understanding of the concepts involved.

This book may be used as a textbook for a one semester course lasting fourteen weeks or longer. This book is intended to be self-sufficient such that readers can study it independently without taking another formal course. However, there are some prerequisites before starting this book, which include a course on engineering mathematics and an introductory course on solid mechanics.

## Inhaltsverzeichnis

### Chapter 1. Introduction

The following are foundational assumptions for continuum mechanics :
Kwansoo Chung, Myoung-Gyu Lee

### Chapter 2. Plasticity Characteristics (in Simple Tension/Compression)

As discussed in Chapter 1, material properties, or more specifically mechanical properties, are required in addition to Newton’s laws to solve the deformation of materials under external forces in continuum mechanics. However, mechanical properties that address all the relationships between stress and strain measures under various conditions are so diverse that measuring them, even only partially, remains as one of the most challenging technical areas.
Kwansoo Chung, Myoung-Gyu Lee

### Chapter 3. Instability in Simple Tension Test

As discussed in Chap. 2, the UTS (ultimate tensile strength) point observed in the simple tension test for both sheet and bulk specimens is important as it is the limit of uniform deformation in the gauge length, which is analyzed here.
Kwansoo Chung, Myoung-Gyu Lee

### Chapter 4. Physical Plasticity

Physical plasticity deals with issues relevant to plastic deformation in the microstructural level, which is therefore beyond the scope of continuum plasticity. However, a few basic features are briefly reviewed here, since these provide some theoretical foundations of continuum plasticity, as will be discussed later.
Kwansoo Chung, Myoung-Gyu Lee

### Chapter 5. Deformation of Heterogeneous Structures

As previously discussed, plastic deformation occurs by dislocation sliding or twinning, driven by shear stress; however, dislocation sliding is predominant at room temperature for most metals with a few exceptions. As for the shear stress to induce the plastic deformation, known as the critical shear stress , its true magnitude is much lower than the theoretical value, with sliding facilitated by dislocations, on a single crystal level.
Kwansoo Chung, Myoung-Gyu Lee

### Chapter 6. Pure Bending and Beam Theory

The deflection of a beam , defined as a uniform long straight slender bar under transverse loading, is discussed here.
Kwansoo Chung, Myoung-Gyu Lee

### Chapter 7. Torsion

Torsion of a cylindrical shaft is an important engineering problem especially since it has the exact analytical solution of the linear isotropic elasticity with infinitesimal deformation for a circular cylinder. The uniform circular cross-section may have an arbitrary size (with the radius of ‘a’ here). Note that the pure bending also has the exact analytical solution of the linear isotropic elasticity but its object may have an arbitrary cross-sectional shape unlike the case of torsion here, which is only for circular cross-sections. The infinitesimal elastic solution is extended here for plasticity with finite deformation considering the one-dimensional elasto-perfect plasticity as a first order approximate solution.
Kwansoo Chung, Myoung-Gyu Lee

### Chapter 8. Stress

In continuum mechanics, an element of a whole body has a mass (dm: the differential mass) and a volume (dV: the differential volume) as well as a shape. The shape is typically considered to be a hexahedron whose six surfaces are aligned with the coordinate system as shown in Fig. 8.1. The coordinate system in this whole book is the rectangular Cartesian coordinate system, which is denoted as x-y-z or 1-2-3 (for the indicial notation), with unit base vectors, $${\mathbf{e}}_{x} ( = {\mathbf{e}}_{1} )$$, $${\mathbf{e}}_{y} ( = {\mathbf{e}}_{2} )$$ and $${\mathbf{e}}_{z} ( = {\mathbf{e}}_{3} )$$.
Kwansoo Chung, Myoung-Gyu Lee

### Chapter 9. Tensors

The stress with nine components derived in Eq. (8.​1) is a tensor. The main task of tensors is to transform one vector to another; i.e.,
Kwansoo Chung, Myoung-Gyu Lee

### Chapter 10. Gradient, Divergence and Curl

The differential operator Nabla, $$\nabla$$, is defined as .
Kwansoo Chung, Myoung-Gyu Lee

### Chapter 11. Kinematics and Strain

Consider the changes in position and shape of a continuum body with time, t, as shown in Fig. 11.1. Then,
Kwansoo Chung, Myoung-Gyu Lee

### Chapter 12. Yield Function

In the simple tension test, materials deform elastically until stress reaches the yield point, after which plastic deformation starts as schematically shown in Fig. 2.​2. Since there are nine stress components (or six components, if its symmetry is considered), combined loading of some or all of those components forms a yield surface, which defines a boundary of elasticity.
Kwansoo Chung, Myoung-Gyu Lee

### Chapter 13. Normality Rule for Plastic Deformation

As for plastic deformation, in order to account for the deformation path (or history) dependent on mechanical properties in plasticity, the plastic (natural) strain increment, $$d{\varvec{\upvarepsilon}}^{p} ( {=} {\mathbf{D}}^{p} dt)$$, is extensively applied as discussed in Remark #11.4.
Kwansoo Chung, Myoung-Gyu Lee

### Chapter 14. Plane Stress State for Sheets

When plasticity is applied to thin sheets such as membranes, plates and shells, the yield function and the plastic strain increment function as well as their applications to the dual normality rules become simpler.
Kwansoo Chung, Myoung-Gyu Lee

### Chapter 15. Hardening Law for Evolution of Yield Surface

In the past few decades, a few experimentations have been conducted to better understand the evolution of the yield surface during plastic deformation.
Kwansoo Chung, Myoung-Gyu Lee

### Chapter 16. Stress Update Formulation

The constitutive law of plasticity consists of three elements: the yield surface defined by the yield function to describe the elasticity limit, the normality rule to define the directions of plastic deformation (for elasto-plasticity) or the stress (for rigid-plasticity) and hardening behavior to describe the yield surface evolution during plastic deformation.
Kwansoo Chung, Myoung-Gyu Lee

### Chapter 17. Formability and Sprinback of Sheets

The main applications of metal plasticity include the process analysis and design of metal forming, which broadly consists of sheet forming and bulk forming.
Kwansoo Chung, Myoung-Gyu Lee

### Backmatter

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